We study the hardness of approximating two reconfiguration problems. One problem is Maxmin $k$-Cut Reconfiguration, which is a reconfiguration analogue of Max $k$-Cut. The other is Maxmin E$k$-SAT Reconfiguration, which is a reconfiguration analogue of Max E$k$-SAT. The Probabilistically Checkable Reconfiguration Proof theorem due to Hirahara and Ohsaka (STOC 2024) and Karthik C. S. and Manurangsi (2023) implies that Maxmin 4-Cut Reconfiguration and Maxmin E3-SAT Reconfiguration are PSPACE-hard to approximate within a constant factor. However, the asymptotic behavior of approximability for these problems with respect to $k$ is not well understood. In this paper, we present the following hardness-of-approximation results and approximation algorithms for Maxmin $k$-Cut Reconfiguration and Maxmin E$k$-SAT Reconfiguration: $\bullet$ For every $k \geq 2$, Maxmin $k$-Cut Reconfiguration is PSPACE-hard to approximate within a factor of $1 - \Omega\left(\frac{1}{k}\right)$, whereas it can be approximated within a factor of $1-\frac{2}{k}$. Our lower and upper bounds demonstrate that Maxmin $k$-Cut Reconfiguration exhibits the asymptotically same approximability as Max $k$-Cut. $\bullet$ For every $k \geq 3$, Maxmin E$k$-SAT Reconfiguration is PSPACE-hard (resp. NP-hard) to approximate within a factor of $1-\Omega\left(\frac{1}{9^{\sqrt{k}}}\right)$ (resp. $1-\frac{1}{8k}$). On the other hand, it admits a deterministic $\left(1-\frac{2.5}{k}\right)$-factor approximation algorithm, implying that Maxmin E$k$-SAT Reconfiguration displays an asymptotically approximation threshold different from Max E$k$-SAT.

翻译：我们研究了两个重配置问题的近似难度。其一是Maxmin $k$-Cut Reconfiguration，它是Max $k$-Cut的重配置版本；另一是Maxmin E$k$-SAT Reconfiguration，它是Max E$k$-SAT的重配置版本。基于Hirahara与Ohsaka（STOC 2024）以及Karthik C. S.与Manurangsi（2023）的概率可检查重配置证明定理，Maxmin 4-Cut Reconfiguration和Maxmin E3-SAT Reconfiguration在常数因子内的近似是PSPACE难的。然而，这些问题的可近似性关于$k$的渐近行为尚未得到充分理解。本文针对Maxmin $k$-Cut Reconfiguration和Maxmin E$k$-SAT Reconfiguration，提出以下近似难度结果与近似算法：$\bullet$ 对于任意$k \geq 2$，Maxmin $k$-Cut Reconfiguration在因子$1 - \Omega\left(\frac{1}{k}\right)$内的近似是PSPACE难的，而它可以在因子$1-\frac{2}{k}$内被近似。我们的下界与上界表明，Maxmin $k$-Cut Reconfiguration展现出与Max $k$-Cut渐近相同的可近似性。$\bullet$ 对于任意$k \geq 3$，Maxmin E$k$-SAT Reconfiguration在因子$1-\Omega\left(\frac{1}{9^{\sqrt{k}}}\right)$（对应PSPACE难）或$1-\frac{1}{8k}$（对应NP难）内的近似是困难的。另一方面，该问题存在一个确定性的$\left(1-\frac{2.5}{k}\right)$因子近似算法，这意味着Maxmin E$k$-SAT Reconfiguration呈现出与Max E$k$-SAT不同的渐近近似阈值。